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Theorem eltrrels2 37962
Description: Element of the class of transitive relations. (Contributed by Peter Mazsa, 22-Aug-2021.)
Assertion
Ref Expression
eltrrels2 (𝑅 ∈ TrRels ↔ ((𝑅𝑅) ⊆ 𝑅𝑅 ∈ Rels ))

Proof of Theorem eltrrels2
Dummy variable 𝑟 is distinct from all other variables.
StepHypRef Expression
1 dftrrels2 37958 . 2 TrRels = {𝑟 ∈ Rels ∣ (𝑟𝑟) ⊆ 𝑟}
2 coideq 37626 . . 3 (𝑟 = 𝑅 → (𝑟𝑟) = (𝑅𝑅))
3 id 22 . . 3 (𝑟 = 𝑅𝑟 = 𝑅)
42, 3sseq12d 4010 . 2 (𝑟 = 𝑅 → ((𝑟𝑟) ⊆ 𝑟 ↔ (𝑅𝑅) ⊆ 𝑅))
51, 4rabeqel 37635 1 (𝑅 ∈ TrRels ↔ ((𝑅𝑅) ⊆ 𝑅𝑅 ∈ Rels ))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 395   = wceq 1533  wcel 2098  wss 3943  ccom 5673   Rels crels 37558   TrRels ctrrels 37570
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-rels 37868  df-ssr 37881  df-trs 37955  df-trrels 37956
This theorem is referenced by:  eltrrelsrel  37964
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